## Abstract

Given two codes R and C, their tensor product R⊗C consists of all matrices whose rows are codewords of R and whose columns are codewords of C. The product R ⊗C is said to be robust if for every matrix M that is far from R⊗C it holds that the rows and columns of M are far on average from R and C respectively. Ben-Sasson and Sudan (RSA 28(4), 2006) have asked under which conditions the product R⊗C is robust. Addressing this question, Paul Valiant (APPROX-RANDOM 2005) constructed two linear codes with constant relative distance whose tensor product is not robust. However, one of those codes has a sub-constant rate. We show that this construction can be modified such that both codes have both constant rate and constant relative distance. We also provide an alternative proof for the non-robustness of the tensor product of those codes, based on the inverse direction of the "rectangle method" that was presented by the second author (ECCC TR07-061). We believe that this proof gives an additional intuition for why this construction works.

Original language | English |
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Pages (from-to) | 351-355 |

Number of pages | 5 |

Journal | Information Processing Letters |

Volume | 112 |

Issue number | 8-9 |

DOIs | |

State | Published - 30 Apr 2012 |

Externally published | Yes |

### Bibliographical note

Funding Information:✩ A preliminary version of this paper was published as ECCC TR07-062. This research was partially supported by the Israel Science Foundation (grant No. 460/05). * Corresponding author. E-mail addresses: [email protected] (O. Goldreich), [email protected] (O. Meir).

## Keywords

- Locally testable codes
- Product code
- Robust
- Robustness
- Tensor product
- Theory of computation

## ASJC Scopus subject areas

- Theoretical Computer Science
- Signal Processing
- Information Systems
- Computer Science Applications